The Stacks project

Remark 43.15.6 (Trivial generalization). Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $M$ be a finite $A$-module. Let $I \subset A$ be an ideal. The following are equivalent

  1. $I' = I + \text{Ann}(M)$ is an ideal of definition (Algebra, Definition 10.59.1),

  2. the image $\overline{I}$ of $I$ in $\overline{A} = A/\text{Ann}(M)$ is an ideal of definition,

  3. $\text{Supp}(M/IM) \subset \{ \mathfrak m\} $,

  4. $\dim (\text{Supp}(M/IM)) \leq 0$, and

  5. $\text{length}_ A(M/IM) < \infty $.

This follows from Algebra, Lemma 10.62.3 (details omitted). If this is the case we have $M/I^ nM = M/(I')^ nM$ for all $n$ and $M/I^ nM = M/\overline{I}^ nM$ for all $n$ if $M$ is viewed as an $\overline{A}$-module. Thus we can define

\[ \chi _{I, M}(n) = \text{length}_ A(M/I^ nM) = \sum \nolimits _{p = 0, \ldots , n - 1} \text{length}_ A(I^ pM/I^{p + 1}M) \]

and we get

\[ \chi _{I, M}(n) = \chi _{I', M}(n) = \chi _{\overline{I}, M}(n) \]

for all $n$ by the equalities above. All the results of Algebra, Section 10.59 and all the results in this section, have analogues in this setting. In particular we can define multiplicities $e_ I(M, d)$ for $d \geq \dim (\text{Supp}(M))$ and we have

\[ \chi _{I, M}(n) \sim e_ I(M, d) \frac{n^ d}{d!} + \text{lower order terms} \]

as in the case where $I$ is an ideal of definition.

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